This almost happened on Apollo 13, when a severe longitudinal oscillation happened on the center engine of the 2nd stage. This was briefly shown in the movie. In the actual mission, later analysis showed the vibration was so severe it nearly caused structural failure.
The second stage thrust structure was apparently rated to 15Gs, which was somewhere between 1/2 and 1/4 of what it experienced during Apollo 13. It's entirely possible that this was actually the closest the Apollo program came to losing a crew after the Apollo 1 fire.
Prior to this flight, POGO was viewed as more of a problem for astronaut comfort and their ability to function than as a potential structural failure issue, which is why the Saturn engineers were willing to live with it. After Apollo 13, they put it a POGO suppressor and the problem did not re-occur. The Shuttle was designed with POGO suppression from the get-go.